Differential Equations

A differential equation is an equation that involves derivatives, or differentials. We previously look the differential equation of population growth. \[\frac{dP}{dt} = kP\] or \[P^\prime(t) = k \cdot P(t)\] where \(P\), or \(P(t)\) is the population at time \(t\). This equation is a model of uninhibited population growth. Its solution is the function \[P(t) = P_0e^{kt}\] where the constant \(P_0\) is the size of the population at \(t=0\). In general, differential equations have far reaching applications and have solutions that are functions. …

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Probability: Expected Value & Normal Distribution

Let \(x\) be a continuous random variable over the interval \([a, b]\) with probability density function \(f\). The expected value of \(x\) is defined by \[E(x) = \int_a^b x \cdot f(x)dx\] The Wikipedia’s definition of expected value is useful in this case. The expected value of a discrete random variable is the probability-weighted average of all its possible values. In other words, each possible value the random variable can assume is multiplied by its probability of occurring, and the resulting products are summed to produce the expected value. …

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Probability

A number between 0 and 1 that represents the likelihood that an event will occur is referre to as the event’s probability. A probability of 0 means that the event is impossible (will never occur), and a probability of 1 means that the event is certain to occur. There are two types of probability.Experimental probabilities are determined by making observations and gathering data. Theoretical probabilities are determined by reasoning mathematically. …

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Business & Economics Applications of Growth / Decay Models Integration

from math import e def pv(p, k, t): k = k / 100 return p * e**(-k*t) def fv(p, k, t): k = k / 100 return p * e**(k*t) def apv(r, k, t): k = k / 100 return (r / k) * (1 - e**(-k*t)) def afv(r, k, t): k = k / 100 return (r / k) * (e**(k*t) - 1) Present Value of a Trust In 18 years, Maggie Oaks is to receive $200,000 under the terms of a trust established by her grandparents. …

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Improper Integrals

An improper integral is an integral that has infinity in one of its “extremities”. \[\int_a^\infty f(x)dx = \lim_{b \to \infty} \int_a^b f(x)dx\] This means that the upper limit of this integral is the limit that \(b\) approaches as \(b\) goes to infinity. If the limit exists, we say tha the improper integral converges, otherwise, we say that it diverges. Two more type of improper integrals follow from the first definition. …

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